laden the scale pan should be moved along the scale beam until it reaches a point where the weights are in equilibrio. The reading of the scale beam gives the average date of the purchases. The entire process requires so little thought that one can determine a large number of averages without fatigue, and can verify the work by simply glancing a second time over the platform and pan for re-assurance as to the proper location of the weights. After the solution of any problem the weights are dumped and screened, thus preparing the machine for the next solution. It is curious to note that the same machine can be used for solving a great variety in simple and inverse proportion, besides many others of a more intricate character, all of which may be determined with the greatest rapidity. RADIO-DYNAMICS. II. By PLINY EARLE CHASE, LL.D. Abstract of Lectures delivered before the Franklin Institute, March 10 and 17, 1881. Continued from page 65. We have confined ourselves thus far to the simple dynamic laws of radial action to or from centres of energy and of inertia. I hope to satisfy you that those laws are sufficient for the explanation of all varieties of physical phenomena, and that radio-dynamics is, therefore, the foundation of all dynamics, the UNIVERSAL physical science, of which photo-dynamics, thermo-dynamics, electro-dynamics, cosmodynamics, moleculo-dynamics and chemistry are branches. Following the Baconian method, I will first lay before you some general principles, followed by some additional FACTS to which I have been led by applying mathematical principles to the study of radio-dynamies, and then interpret those facts, in the hope of making them helpful towards the strengthening of hypotheses and the discovering of new facts or laws. The elements of physical energy are mass, m, and velocity, v. The total energy of any force is called its ris riva, or living force. It is measured by the work that it is able to accomplish against uniform m 1,2 resistance, and is represented by 2 La Place defined velocity as "the ratio of the space to the time employed in describing it." In recent dynamical treatises the space traversed, or length, is represented by ; time of traverse, by t; velom 12 12 city, by; energy, by t The greatest or controlling energy must, of course, be represented by the product of the greatest mass by the square of the greatest velocity. The greatest mass of which we have any practical knowledge is the mass of the Sun, M; the greatest centripetal acceleration is the force of gravity at Sun's surface, G.; the greatest velocity of wave propagation is the velocity of light, or the velocity of radiation, I, at the seat of greatest centripetal acceleration. Action and reaction being equal and in opposite directions, we may reasonably look for some simple relation between the centripetal and centrifugal maxima, G, and V%. If we regard Sun's radiating energy as an action, its reaction must be dependent upon its inertia, or mass. The velocity of projection against uniform gravitating resistance is represented by gt, t being one-half the time of flight. The height of projecgt2 tion is The height of projection which would give the velocity 2 of wave propagation in an elastic medium is one-half the "height of a homogeneous atmosphere" of the medium. Every particle of the Sun's surface is continually solicited by gravitating tendencies, G, and G, towards its own centre and towards the centre of the universe. In each solar rotation the particles are alternately projected from and drawn towards the centre of the universe, Cu = We may, therefore, let T, = solar rotation, represent the time of cyclical equality of action and reaction between solar inertia and universal inertia, or between solar gravitation and æthereal undulation. Then L G T is the height of a homogeneous æthereal atmosphere, at Sun's surface, which would have a velocity of wave propagation equivalent to the velocity of light ; = is the velocity of T. light. 0 All forms of energy, mechanical, thermal, photic, electric, magnetic or chemical, are derived from and can be compared with the maxi 1. All astronomical, barometric or other mechanical estimates of solar mass and distance involve the proportionality, g m 12 2. The arbitrary units of thermo-dynamics are based upon the work done against the centripetal accelerations of superficial terrestrial gravity. The unit of acceleration, or the sum of accelerations in unit of time, at unit of distance, is proportioned to mass. Therefore, if we designate Earth's mass by mg, we have the proportion 0 I is the velocity acquired during the cyclical actions and reactions of solar condensation and ethereal elasticity, at Sun's surface; ug = 56558 mile is the velocity acquired during the cyclical actions and reactions of water congelation and vaporization at Earth's surface. Dividing by 1 180 for the Fahrenheit scale, or by 1 100 for the Centigrade scale, we have the arbitrary units of velocity, 042156 mile for 1°F., 056558 mile for 1°C. The equation V1 ́2gh gives h= 772 ft. for 1°F., or 1389.6 ft., = 424 metres, for 1°C. Combining these heights with the arbitrary units of mass, we have J, = 772 ft. lbs., for the English thermal unit, and C, = 424 kilogrammetres, for the calorie or French thermal unit. be com 3. Thermal, mechanical and photo-dynamic energies may pared with energies of chemical combination, through the ratios. Mom3 V6: Uz :: ho he 0 Earth's mean distance from Sun, or height of solar projection, is represented by h; h is 4 of 1⁄2 of the height to which water vapor would be thrown, against the retardation of gravity, by the combining energy of oxygen and hydrogen, the two constituents of water; is the length of a conical pendulum which would vibrate in the same time as a linear pendulum of unit length; is the ratio of ris vira of wave propagation to the mean vis viva of the oscillating particles which originate the waves. 4. Cosmical, electrical and photo-dynamic energies may be compared by means of the ratio M Vo v3 mg V2 :: m2 V2: mz Vo vz 0 0 Sun, M, is at the centre of nucleation in the solar system; Earth, m, at the centre of condensation; Jupiter, m,, at the nebular centre; is Earth's aphelion or "nascent" orbital velocity. 5. The electro-static, magnetic and electro-kinetic units of energy can all be derived from the above expression for Earth's photo-dynathrough the equations [e E] = [mL] mic energy, m3 V6 = [p C] mz = 0 M3 L2 The bracketed symbols represent, respectively, quantity of electricity; line integral of electromotive force, or electric potential; quantity of free magnetism, or strength of a pole; magnetic potential; electro-kinetic momentum of a circuit; electric current. 6. Atomic energy, or energy of unit volume, can be compared with Earth's photo-dynamic energy of unit volume, L3 m3 L2 2 133 T2 73 with corresponding electric energies, through the equations [DE]=[B H]=[CV] = M M3 and The bracketed symbols represent, respectively, electric displacement (measured by surface density); electromotive force at a point; magnetic induction; magnetic force; current electric intensity at a point; vector potential of electric current. 7. Electro-chemical and electro-magnetic energies may be compared with thermal, photo-dynamic and other energies through the propor I designate Weber's units of electro-chemical and electro-magnetic force by z and u, respectively; t is the time of acquiring orbital velocity, or incipient associative energy, at Laplace's limit of equal velocities of rotation and revolution; t is the time of acquiring nucleal nascent or dissociative velocity. The ratio of t to t is the same as the ratio of the diameter of a circle to its circumference, 1 π. n 8. Total magnetic force, 4, can be compared with the reactions of terrestrial magnetic force, 3, by the proportion a The reactions of orbital tendency are t M, t m', respectively. Centripetal undulation varying as the fourth power of orbital velocity, we have the ratio t M tm, or M2 : z1 m22. 9. In the actions and reactions between the centre of nucleation, M, and the centre of condensation, my, there are continual tendencies towards the centre of gravity, the centre of lineal oscillation and the centre of conical oscillation, as well as to centripetal and orbital motions. These are all satisfied by the proportion M2 mz (2×3×4)' : 1' :: 331776 : 1 In order to show the closeness of accordance between the theoretical results and those which have been deduced from observation or experiment, I make the following comparisons: 1. The theoretical oscillatory value of Sun's mass, (9), gives 92,785,700 miles for Sun's distance; 8.809" for Sun's parallax; 25-496 days for Sun's rotation; 299,943 kilometres per second for the velocity of light. These values differ by less than of one per cent. from Fave's, Michelson's and La Place's estimates. 2. The difference between the theoretical and experimental thermodynamic velocities is less than of one per cent. Perhaps, when proper allowances are made for the temperature at which the experiments were performed, the accordance will be found to be exact. 3. The difference in the combining energy of hydrogen and oxygen is less than of one per cent. 4. The difference in Earth's "nascent " velocity is less than 1 of one per cent. 5, 6. The difference which is indicated by the latest electro-dynamic and electro-magnetic investigations is less than 5 of one per cent. 7. Weber's experimental determination of the electro-chemical unit was of one per cent. greater than the theoretical value. 8. The experiments of Joule and others, in Great Britain, upon magnetic action and terrestrial reaction, indicate a difference of about 1 per cent. in the estimate of Sun's mass, and of one per cent. in the estimate of Sun's distance. I have been obliged to use many technical terms and equations which are doubtless new to most of you, but you can all see how close is the agreement of the values which are derived mathematically from simple radio-dynamic action, with those which have been obtained experimentally by the most careful observers. You will have noticed how often I have called your attention to evidences of the fourth fundamental law, which is generally known as "Fourier's Theorem," viz.; every periodic vibrating or orbital motion can be regarded as the sum of a certain number of pendulum vibrations. To illustrate this law I have prepared a stretched cord, to which I |